Unitary interpolation for operators in tridiagonal algebras

Abstract

Given operators $X$ and $Y$ acting on a Hilbert space $\Cal H$, an interpolating operator is a bounded operator $A$ such that $AX=Y$. An interpolating operator for the $n$-operators satisfies the equation $AX_i=Y_i$, for $i=1,2,\cdots,n$. In this article, we obtained the following : Let $X = (x_{ij})$ and $Y =(y_{ij})$ be operators acting on $\Cal H$ such that $x_{i \sigma(i)} \neq 0$ for all $i$. Then the following statements are equivalent. $$ \aligned \text{\rm (1)} & \ \text{\rm There exists a unitary operator $A$ in Alg$\Cal L$ such that $AX =Y$} \\ & \ \text{\rm and every $E$ in $\Cal L$ reduces $A$.} \\ \text{\rm (2)} & \ \text{\rm $\displaystyle \sup \left\{ {{\| \sum_{i=1}^n E_i Y f_i\|} \over{\| \sum_{i=1}^n E_i X f_i\|}} : n \in N, E_i \in {\Cal L} \ \text{\rm and} \ f_i \in {\Cal H} \right\} < \infty$ and } \\ & \ \text{\rm ${|y_{i \sigma(i)}| \over |x_{i \sigma(i)}|} = 1$ for all $i=1,2,\cdots$.} \endaligned $$